Curve Fitting to Planar Data

在本课题中，我们首先借助计算机数学系统MATHEMICA得到了参数有理三次曲线线段上的拐点和奇点的分布。末端斜率大小的倒数决定了线段上拐点和奇点的出现，它的使用使我们能够检查线段是否有拐点或奇点(环状或尖点)，以及如何放置控制点和如何为有理Bezier三次曲线线段选择权重以保持光滑的形状。缓和曲线段有几个优点，既不包含拐点、奇点，也不包含曲率极值。然后，我们给出了(I)平面T-三次曲线段由末端斜率的倒数表示为螺线的一个容易使用的条件，(Ii)T-三次螺线的显式形式，(Iii)形成T-三次螺线和圆弧/T-三次螺线的简单算法。我们还讨论了根据数据点的切线向量的角度应该使用哪些螺线。这些结果对于生成一组平面数据点的平面“视觉愉悦”、“保形”近似是有用的。

In this project, first we have obtained the distribution of inflection points and singularities on a parametric rational cubic curve segment with aid of Mathematica(A System for Doing Mathematics by Computer). The reciprocal numbers of the magnitudes of the end slopes determine the occurrence of inflection points and singularities on the segment Its use enables us to check whether the segment has inflection points ora singularity (a loop or a cusp) and to get an idea how to place control vertices and how to choose weights for the rational Bezier cubic curve segment to preserve the fair shape. Spiral segments have several advantages of containing neither inflection points, singularities nor curvature extrema. Next, we have given (i) an easy to use condition for a planar T- cubic .segment to be a spiral in terms of the reciprocal numbers of the magnitudes of the end slopes, (ii) the explicit form of the T-cubic spiral, and (iii) simple algorithms for forming a T-cubic spiral and an arc/T-cubic spiral. We have also discussed which spirals should be used according to the angles of the tangent vectors at the data points. These results are useful for generating planar "visually pleasing", "shape preserving" approximations to a set of planar data points.